Pacific Journal of
Mathematics
THE FACTORIZATION OF H p ON THE SPACE OF
HOMOGENEOUS TYPE
A KIHITO U CHIYAMA
Vol. 92, No. 2
February 1981
PACIFIC JOURNAL OF MATHEMATICS
Vol. 92, No. 2, 1981
p
THE FACTORIZATION OF H ON THE SPACE OF
HOMOGENEOUS TYPE
AKIHITO UCHIYAMA
r
Let A be a Calderon-Zygmund singular integral operator
with smooth kernel. That is, there is an Ω(x) defined on
n
R \{0} which satisfies
ί
Ω= 0 , Ω m0 ,
Jl»l=t
(* )
n
Ω(rx) = Ω(x) w h e n
\Ω{x)-Ω(y)\<\x-y\
r > 0 a n d x e R \{0} ,
w h e n \χ\ = \y\ = l,
and that
n
Kf(x) = P.V.[
Ω{x-y)\x-y\- f(y)dy.
K'f(x) = P. F. (
β(y - x)\y - x \~ Ay)dy .
Let
n
R. Coifman, R. Rochberg and G. Weiss showed the weak verx
n
sion of the factorization theorem of H (R ) and that was
refined by Uchiyama in the following form.
THEOREM A.
// 1 < q < ^
CKJI/WHHR*)
^
and 1/q + 1/r = 1, then
inf
/ = Σ.(.htKgt - gtK%)\
i =l
)
£ c'κ
In this note, we extend Theorem A to HV(X), where p e (1 — sx, 1]
and X is a space of homogeneous type with certain assumptions.
1* Preliminaries* In the following, A > 1 and 7<;i are positive
constants depending only on the space X
Let X be a topological space endowed with a Borel measure μ
and a quasi-distance d such that
(1)
( 2)
d(x,
y)^0
d(x, y) > 0 iff
xΦ y
(3)
d(x, y) = d(y, x)
(4)
d(x, z) £ A{d{x, y) 4- d(y, z))
453
454
AKIHITO UCHIYAMA
I d(x, z) - d(x, y) \/d(x, y) £ A(d(z, y)/d(x, y))r
if d(z, y) < d(x, y)/(2A)
( 6)
t/A£ μ(B(x, t)) ^ t
for all x, y, z in X and all t e (0, Aμ(X)), where B(x, t) = {y e X:
d(x, y) < i). We postulate that {B(x, t)}te(0, Aμ{x)) form a basis of open
neighborhoods of the point x.
Let φ{t) e C°°(0, w) be a fixed nonnegative function such that
φ(t) = 0 on (0, 1/2), φ{t) = 1 on (1, oo) and |d^/dί| < 3.
Further, we assume that X is endowed with a function k(x, y)
defined on X x X such that
(7)
\k(x,y)\^l/d(x,y)
( 8)
for all a?, y e X
sup{|fc(α, y)\:yeX
satisfying A~2ί ^ d(a;, T/) ^ t} ^
sup{| Jc(y, x)\:yeX
satisfying A^t ^ d(x, y) ^ t} ^
for all x G X and all ί e (0, Aμ{X))
9
i ^ , 1/) ~ k(%, «)l, 1^(2/, a?) - &(s, ») I ^ (d(2/, z)/d(x, y))r/d(x, y)
if d(2/, z) < d(x, y)/(2A)
2
and that for any / e L (X)
Kf(x) = lim^fo Jfc(ί», i/, t)f(y)dμ(y)'
K'f(x) = lim^+o Jfc(i/, x, t)f(y)dμ(y)
exist almost everywhere and
(10)
!|iΓ/||2^||/||2, | | ί Γ 7 | | 2 ^ | | / | | 8 >
where
fc(a, 2/, ί) = k(x, y)φ(d{x, y)/t)
G
\l/P
χ
\9(y)\pdμ(v)J
.
For xeX and ί e (0, Aμ(X)), let
T(x, ί) = {ΨeC(X):
(11)
supp f c B(x, t)
(12)
iy(ί/)i^i/ί
(13)
I Ψ(y) - Ψ(z) I ^ (d(yf z)/ί)V« for any y,zeX}
For / e L'(X) and j> > 1/(1 + 7), let
/*(*)=
sup sup
ί e (0,/l/<(ΛΊ)
ΨeT(x,t)
.
THE FACTORIZATION OF HP ON THE SPACE OF HOMOGENEOUS TYPE 455
If p > 1, then 11 /1 \HP ^ \\ f \ \p by t h e Hardy-Littlewood maximal theorem
P
P
and we define H (X) = L (X).
If 1/(1 + y)< p ^ 1, then we define
P
H (X) t o be t h e completion of {feL\X):
\\f\\Hp < °°} by t h e metric
11/-A
A comment on notation: The letter C denotes a positive constant
depending only on A and 7. The various uses of C do not all denote
the same constant. All the functions considered are real valued
functions.
2* The results* Our results are the following.
THEOREM 1. // 1/p = 1/q + 1/r < 1 + 7, 0 < 1/q < 1 + T, 0 < 1/r <
1 + 7, g e Hq n L2 and h e Hr Π L\ then
\\hKg
where cq>r is a positive constant depending on q, r and X.
REMARK 1. As a consequence of this theorem, for any geH9
r
and any heH we can define hKg — gK'h as the limit of [hjίg^ —
gjK'hj}™^ in Hp, where {f^JJU aHq f) L2 converges to g in Hq and
{hj}™^ c Hr Π L2 converges to h in Hr.
THEOREM 2. If μ(X) = oo, l ^ l/p = ι/q + l/r < 1 + 7, 0 ^ 1/g <
1 + 7, 0 ^ 1/r < 1 + 7 and f e Hp, then there exist {gj}f=1 c Hq and
{hj}f=1c:Hr such that
/ = Σ (hdκgj - gόK%) ,
As a result of these theorems, we get
COROLLARY 1. // μ(X) = «>, l ^ i/ p = i/ g + i/ r < i + 7 , o <
1/q < 1 + 7, 0 < 1/r < 1 + 7 cmd / e i P ,
β f i r | | / I U , ^ i n f {(gdlflryllπtllλyl
/ =Σ
ii
EXAMPLE 1. LetX=R% d(x, y) = \x-y\nωn=(ΣΛU(xJ~yjyr/2ωnf
μ
be the Lebesgue measure and let k(x, y) — Ω(x — y)\x — y\~n, where
n
ωn is the volume of the unit ball of R and Ω satisfies (*). Then,
by taking 7 = 1/n and by taking A sufficiently large depending on
456
AKIHITO UCHIYAMA
Ω, the conditions (1) ~ (10) can be satisfied. In this case, the above
definition of H1 coincides with the definition of H\Rn) given by
Fefferman-Stein [5]. Thus, Corollary 1 is an extension of Theorem A.
2. Let At = tp(0 < t < oo) be a group of linear transn
formation on R with infinitesimal generator P satisfying (Px, x) ^
n
n
(x, x), where ( , ) is the usual inner product in R . For each xeR
let p(x) denote the unique t such that \A^x\ — 1. Let Ω(x) be such
that
EXAMPLE
(
Jlαl=l
Ω(x)(Pχf
x) = 0 ,
Ω =£ 0 ,
Ω(Atx) = Ω(x) w h e n t > 0 a n d x 6 Rn\{0}
\Ω(x) -Ω{y)\
£ \x-y\
w h e n \χ\ =
\y\=l.
Let X = Rn, d(x, y) ~ p(x — y)ua)n, μ be the Lebesgue measure and
let k(x, y) = i2(a; — y)/d(x, y), where v — tr P. Then, by taking T =
1/v and by taking A sufficiently large depending on P and Ω, the
conditions (1) — (10) can be satisfied. [See Riviere [12].]
If we remove the condition μ{X) = ^°, we can show the following
a little weaker result.
2'. // μ(X) < oo9 x is connected, 1 ^1/p = 1/g + 1/r <
2
1 + 7, 1 < q, 1 < r, / G ίί ' α^ώ [fdμ = 0, ίAew ίλere exist {g^ c
THEOREM
=
±(hjKgj-g3-K'hj)
Γ. // μ{X) < <χ>, Xis connected, l<*l/p = Ijq + 1/r <
q < o o , K r < co, feHp and [fdμ = 0,
COROLLARY
1+7, K
Σ
REMARK
2. When μ(X) < oo, for feL\X)
we can easily show
\\fdμ ^ C inf f*(x) .
p
Thus, for any f eH we can define l/eϊμ by lim \/%cί/^, where {fn} c
THE FACTORIZATION OF W ON THE SPACE OF HOMOGENEOUS TYPE 457
1
p
p
L Π H and \imfn = / in H . And it follows easily that
3* The basic lemmas.
DEFINITION 1. If 1/(1 + 7) < p ^ 1, we say that a function a{y)
is a p-atom if there exists a ball B(x, t) such that
supp a c £(&, ί), I N L ^ ί~1/p, ( αώμ = 0 .
(20)
We can show easily that ||a||#p ^ cp.
DEFINITION 2.
1
For f eL
2
+ L , q > 0 and α > 0, let
Jlfff/(a0 = sup (ί
ί>0
\f\gdμltX/9
\Jΰ(a;,t)
/
ΛΓ*/(») = sup I \k(x, y, t)f{y)dμ{y)
ί>0
I J
K'*f(x) = sup I U(^ f α?, t)f{y)dμ{y)
t>o I J
/*M(cc) = sup sup I [fWdμ
where
Γβ(a;, ί) = {¥ 6 C(X): | f(«) | ^ f(t + d(«, z))" 1 ^
l ^ ^ l ^ φ ) ^ ^
if d{z, y) < ad(x, z)} .
LEMMA
1. If p > q, then
This is an immediate consequence of the Hardy-Littlewood maximal
theorem. We omit the proof.
LEMMA
2. If d{y, z) ^ d(x, y)/(2A), then
d(x, y)/(2A) ^ d(x, z) ^ 2Ad(x, y) .
This follows easily from (4). We omit the proof.
LEMMA
3. If t > 0 and if d(y, z) sΞ d(x, y)/(2A), then
I ψ{d(x, y)/t) - φ(d(x, z)/t) I = 0 if d(x, y) $ (ί/(4A), 2At) ,
g C(d(y, z)/d(x, y)Y
458
AKIHITO UCHIYAMA
otherwise.
Proof. Set w = φ(d(x, y)/t) - <p(d(x, z)/t). If d(x, y) £ ί/(4A),
then, by Lemma 2, d(x, z) S t/2. Thus, w = 0 - 0 = 0. If d(x, y) ^
2Atf then, by Lemma 2, d(x, z) ^ t. Thus, w = 1-1 = 0. If ί/(4A) <
d(x, y) < 2At, then, by (5),
\w\£
LEMMA
C\d(x, y) - d(x, z)\/t ^
4. If t > 0 and if d(y, z) <; d(a;, y)/(2A), then
\k(x, y, t) — k(x, z,t) <i Cd(y, z)rd(x, y)~ι~τ
Ί
l
T
\k(y> x, t) — k(z, x, t) <^ Cd(y, z) d(x, y)~ ~ .
Proof.
We show only the first inequality.
Note that
\k{x, y, t) - k(x, 2, ί ) | =S \k(x, y) - k(x9 z)\φ{d{x, y)/t)
+ \h(x, Z)\ \φ(d(x, y)/t) - φ(d(x, z)/t)\ .
By (9), the first term of (22) is dominated by d(y, z)rd(x, y)~ι~r. By
Lemma 2, Lemma 3 and (7), the second term of (22) is also dominated
by Cd(y, z)rd(x9 yYv~τ.
5. Let 1/(1 + 7) < p ^ 1 and ueHp.
Then, there exist
a sequence of real numbers {λyl^i and a sequence of p-atoms {aj}j^1
such that
LEMMA
(23)
u(x) = Σ λ. α/^)
in
Hp
when
μ(X) = 00 ,
i=i
u(x) = X Xάaά(x) + \udμ/μ(X)
5=1
in Hp when μ{X) < ^ ,
J
This is the atomic decomposition of HP(X) which was shown by
Macias-Segovia [10].
LEMMA 6. Let 1/(1 + 7) < p ^ 1, u e L\ supp u c B(x0, t) and t e
(0, Aμ{X)).
Then, there exists a sequence of real numbers {λ^JU and
a sequence of p-atoms {aό\f=1 such that
CO
(24)
where
,„
THE FACTORIZATION OF HP ON THE SPACE OF HOMOGENEOUS TYPE 459
λ0 - \udμtυημ{B(x0,
t)) , ao(x) = t-u>Xm,Olt)(x)
and XE denotes the characteristic function
of a measurable set E.
Note that \udμ l/ί <: CmfxeB{XQf2At)u*(x).
Then, applying Lemma
5 to u — λoαo, we get Lemma 6.
LEMMA
7.
Let 1/(1 + 7) < p ^ °°.
(25)
ιιr
Proof
cα]
Then,
i!^^
It can be shown easily that
^ caMJ(x) .
f^\x)
Thus, if p > 1, (25) follows from the Hardy-Littlewood maximal
theorem.
Let 1/(1 + 7) < p ^ 1. Note that if μ(X) < 00, then it is trivial
that WXi^ll ^ CpWX*™^ ^ cp,a. Thus, by Lemma 5, it suffices to
show (25) for a p-atom a(y) satisfying (20). If yeB(x,t/a)c,s
>0
and Ψ e Ta(y, s),
\\a(z)W(z)dμ(z)
=
\\a{z){Ψ{z)-Ψ{x))dμ{z)
S \
t-1/pd(z, xYd(x, yYι-τdμ
JB(x,t)
^ tl-lp+rd(x,
y)-1-? .
Thus,
(26)
a*i«\y) ^ t
If yeB(x, t/a), then
(27)
a*w(
Hence, by (26) and (27),
LEMMA
(28)
(29)
8.
Lei 1/(1 + 7) < p < °°.
Then,
\\K*f\\p^cp\\f\\HP
by (21)
460
AKIHITO UCHIYAMA
Proof. If p > 1, then these follow from the well known argument
about the maximal singular integral.
Let 1/(1 + τ ) < V ^ I- We show only (28).
oo, then it follows easily that
Note that if μ(X) <
Thus, by Lemma 5, it suffices to show (28) for a p-atom a{y) satisfying
(20). If d(x, y) > 2 At and s > 0, then
\k(yf z, s)a{z)dμ{z)
=
γjciy, z, s) - k(y, x, s))a{z)dμ{z)
^ C[
JB(x,t)
d(x} z)rd{x, iή-i-rf-ί'pdμ by Lemma 4
Thus,
K*a{y) ^ Ct1-1'*»d(x y)-1^ .
(30)
On the other hand, since (28) has been known for p = 2, we get
(31)
(
\K*a\pdμ ^ ^'^([l^a^dμY2
^ Cί^^Hαllr ^ C .
Thus, by (30) and (31), we g e t
(32)
LEMMA
9. Let ζ(x, y) he a function
defined on X x X such that
(33)
\ζ(x, y) - ζ(x, z)\ ^ d{y, z)rjd(x, y)
if d(y, z) < d(x, y)/(2A).
(34)
Let u e L2, supp u c B(xQf t), t e (0, Aμ(X))
v(cc) = jζ(£c, y)u{y)dμ{y)
and 1 + 7 > 1/Si > 7.
(ί
(35)
where
M
(
B(xo,2At)
l/s2 — l/s x — 7.
Proof.
If Si > 1, this can be shown in the same way as [13]
THE FACTORIZATION OF Ή* ON THE SPACE OF HOMOGENEOUS TYPE 461
120. Let 1/(1 + 7) < Si ^ 1. Applying Lemma 6 to u{x) and p = s19
we get {Xj}f=o and {αy(α0}Γ=o such that (24). For j = 1, 2, 3
, let
{
(36)
B(xjf tj) =) supp ah tγ ^ ^ 11 aό | U .
For i = 0,1,2, -. ,let
(37)
vά{x) - jζ(a;, y)aά{y)dμ{y) .
Then,
(38)
1
"", t f r - l ί s ^ d { x f
x3))
fori ^ l .
Thus, by (24) and s, ^ 1 < s2,
MrfΣIίl
JB(ίB0,t)
(39)
/
i=0
l
\jB(xo,t)
^^Σlλjl^c^
B{x0Λ
4. Proof of Theorem 1. We may assume q ^ r. Then r > 1.
Let x e X be fixed. Let ί 6 (0, Aμ(X)) and r e T(x, t). Then
-
g{y)K'h{y))dμ{y)
(40)
J
- K(Ψg){y))h(y)dμ{y) .
Set
iy(y, z) = k(y, z)(Ψ(y) - Ψ{z))g{z) .
Note that
Ψ{y)Kg(y) - K{Ψg){y) =ty(y,z)dμ(z) .
Let
(41)
d(x, y) > 16A*t .
Then Ψ(y) = 0. Set
\v(y, z)dμ{z) = -k{y, x) \ψ(z)g(z)dμ(z)
(42)
, ίc) - Λ(», z))Ψ{z)g{z)dμ(z)
462
AKIHITO UCHIYAMA
V, z)g(z)dμ(z) = η,{y) + ηt(y) .
If z 6 supp Ψ, then, by (41),
d(x, z) < d(y, x)/(2A) .
Hence, by (9) and (12),
\ζ2(y, z)\ ^ d(x, y^-ψ-1
(43)
.
If zu z2 e B(x, 2At), then, by (41) and Lemma 2,
d(x, 2X) < d(y, x)l(2A)
and d{zlt z2) < d(y, zί)/(2A)
Hence, by (9), (12) and (13),
(44)
^ Ifc(y,x) - fcfo,
2l
) I I Ψ{zx) - ^(« 2 ) [ + | k(y, zj - k(y, z2) \ | ^(2 2 ) |
^ 0(^(2!, zj/typ-'dix, y)-1^ .
Thus, by (43), (44) and suppζ2(2/, ) c £ ( x , ί),
Ct-rd(x, yy+rζ2(y,
-)eT(x,t).
So,
ι
(45)
1 %(ί/) I ^ Cd(a;, y)- -φg*(x)
.
Let
(46)
d(x, ?/) ^ 1<OAH .
Set
= Ψ(y) jfc(α, «)?>(d(», z)/(βt))g(z)dμ(z)
, z) - k(x, z))φ(d(x,
^ ,
z)(Ψ(y) - Ψ(z))φ'(d(x,
z)l(βt))g(z)dμ(z)
z)/(βt))g(z)dμ(z)Imx,16AH)(y)
(47)
J ; , 2, βt)g(z)dμ(z)
, z)φ'(d(x,
+ Ψ(y) Jζ t (y, z)g{z)dμ(z)
z)l(βt))g{z)dμ{z)l{y)
where β = 128Aβ and φ' = 1 — φ.
Since /3 is sufficiently large, if φ(d(x, z)/(βt)) Φ 0, then
d(x, y) < d(x, z)/(2A) .
THE FACTORIZATION OF H" ON THE SPACE OF HOMOGENEOUS TYPE
463
Hence, by (9),
(48)
\Uv, z)\ £ Ct'(t + d(x, z))~^ .
Let
(49)
d(zlf z2) < d(x, «x)
Set
\UV, «i) - UV, «,) I ^ |A<», 2X) - k(x, Z2)\φ(d(X, zj
k
z
+ 1 (y>
oil
+ (I Kv,
= c« + c42
By (49) and (9),
(51)
Since β is sufficiently large, if φ(d(x, zj/iβt))
Lemma 2,
Φ 0, then, by (46) and
d{x, z1)/(2A) :g d{y, zd •
Hence, by (49) and (9),
ζ42 ^ d(zu
By Lemma 2,
(53)
dfo z,) ^ d(x, «1)
If ζ4S > 0, then, by Lemma 3,
d(x, zύ > βtl(AA) .
So
d(x, y) £ 16A4ί ^ 64A5d(x, z2)/β = d(x, zt)/(2A) .
Thus, by Lemma 2 and (53),
(54)
d(y, z2) ^ d(x, z2)/(2A) ^ d(a;, 21)/(2A)2 .
Hence, by (7), Lemma 3, (53) and (54),
1
(55)
ι
ζ* £ (d(y, z,)- + d(x, z2)- )C(d(zlt
^ Cd{zlt z,yd(x, zj-1-* .
So, by (48), (51), (52) and (55),
r
z2)/d(x, zt))
464
AKIHITO UCHIYAMA
Cζt(y, • ) e Γ M - ! ( i , t ) .
Thus,
(56)
M
y
)
2
\
By (7) and (13),
(57)
\ζ,(y,
If d(zu z2) < d(y, zd/(2A), then by (7), (9) and (13),
£ I k(y, zXnzύ
- Ψ(z*)) I + IKV, zt) - k(y, z2) \ \ ¥(z2) - Ψ(y) |
ι
ι
^ d(y, zx)- t- -rd(zx, z2γ + d(y, z^diz,,
z^r^d^,
yγ
^ Cd(y, zι)-1t-ι-rd{zu z2γ .
(
So, by (57) and (58), Ct1+rζ5(y, z) satisfies the hypothesis of Lemma
9. Note that if z e B(x, 2Aβt),
)/(/3t))flr( •))*(«) ^ Cg*(z) .
(φ'(d(x,
Thus, by Lemma 9, we get
\
4
B(x,lQA t)
where 7 < l/s1 < 1 + 7 and l/s 2 = l/s1 — 7.
By (42), (45), (47) and (56),
\V(V, z)dμ(z) = -Wgdμk(y,
x)φ(d(y, x)/t) +
where
\V*(y)\ ^ Cg*ί{2Λ)'\x)tr(t
+ d(x,
Thus,
/, z)dμ(z)h(y)dμ(y)
(60)
^ C]g*(x)K'*h(x)
+ h*(x)K*g(x)
Since 1/j) = 1/q + 1/r and 1/p < 1 + 7, we can take s2 such that
(61)
1 + 7 > 1/β! > max(l/?, 7), 1/βJ = 1 - s2 > 1/r .
THE FACTORIZATION OF W ON THE SPACE OF HOMOGENEOUS TYPE
465
Then, by (59),
(62)
s(
t
Mv)\μ(v))X\
S cSlMSlg*(x)Mφ(x) .
By (40), (60) and (62), we get
(hKg - gK'h)*(x) ^ C{g*(x)K'*h(x) + h*(x)K*g{x)
All the terms on the right hand side belong to Lp by Lemma 1,
Lemma 7, Lemma 8 and (61).
5. Proof of Theorem 2. By Lemma 5, we may assume that
/ is a j)-atom such that
supp / c B(x0, t), 11 /1U < t-ι/"
and
^fdμ = 0 .
Let q ^ r. Then r > 1.
Let N be a large number depending only on X and p.
by (8), there exists y0 such that
A-2Nt ^ d(x0, y0) ^ Nt, \k(y0, xo)\ > l/(ANt) .
By (9),
inί{\ k(y, x)|: d(«, x0) < t, d(y, y0) < t) > l/(2ANt) .
Let
(70)
h(x) = Z B ( ϊ O l i ) (a;)^.
Then, \K'h(x)\ > C on £(«„, ί). Let
flr(ir) = -f(x)IK'h(x0) .
r
Then, geH',heH
and
1
ιιi; He. ιιftUs, ^ cdb-wm *
= CN .
Set
= f{χ) - (h{x)Kg(x) - g(x)K'h(x))
= f(x)(K'h(x,) - K'h(x))/K'h(x0) - h(x)Kg{x)
— wax) + Wt(x) .
Since supp ^ c 5(a;0, ί) and 11 w,. | L ^ t~1/pN"r, we see that
Then,
466
AKIHITO UCHIYAMA
ί
wfp{x)dμ{x)
2
t-ιN~rP(l
^ I
2
+ d(xQf x)/t)~pdμ(x)
τ
jB(xQ,4A Λ t)
^ cpN~rP+1-p log N .
A similar estimate holds for w2.
Thus,
w?2' + wtpdμ
w*p(x)dμ(x) ^ I
I
2
2
JB(x0AA Nt)
τ
JB(xQ,4A Λ t)
^ cpN'ΪP+1-p
log iV
> 0 as
N
> oo .
Since supp w c B(xQ, 2ANt) and Iwd^ = 0, by taking N
sufficiently
large and applying Lemma 6 to w(x), we get
w(x) =
where {fj}f=1 are p-atoms and
oo
3=1
Hence,
/ - (hKg - gK'h) + Σ λy/y .
5=1
Applying the same argument to each f5 and repeating this process,
we get the desired result.
6* Proof of Theorem 2'* Since μ{X) < oo and X i s connected,
we can easily see that for any ε > 0 and any p-atom a(x)f there
exist {aj{x)}%l such that
and that each aά is a p-atom supported on the ball with radius <ε.
Thus, for the proof of Theorem 2', we may assume that / is a
p-atom such that the radius of its support is less than N~ιμ(X),
where JV is a sufficiently large number, depending only on X and
p, to be determined later.
Following the proof of Theorem 2, we define h(x) by (70) and
g(x) by
g(x) - -f(x)/K'h(x)
Then,
.
THE FACTORIZATION OP W ON THE SPACE OF HOMOGENEOUS TYPE 467
w(x) = f{x) - (h(x)Kg(x) - g(x)K'h{x)) = -h(x)Kg(x) .
Note that if y e B{yQ, t), then
/, z)f(z)dμ(z)/K'h(x0)
/, z)f(z)(l/K'h(x0) - 1/K'h(z))dμ(z)
£c\\k(y,z)-k(y,xo)\\f(z)\dμ(z)
+ \\k{y,z)\\f{z)\N-rdμ{z)
SΞ c
^Ntrw-rifωidμω ^
CN-1-^*
.
Thus,
,||A|| r ^ C||/IU|Λ|| Γ ύ C r ^ W
= CN,
μ = 0,
supp w c B(y0, t)
l | w | L ^ \\h\Usu-pytmyo,t)\Kg(y)\ ^
If iV is sufficiently large, then 2w is a ^-atom and the radius of its
support is less than N^μiX).
Iterating this process, we get desired
result.
I thank Professor M. Kaneko for helpful
conversation and valuable information.
ACKNOWLEDGMENT.
REFERENCES
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distribution, Advances in Math., 16 (1975), 1-64.
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3. R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces
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p
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468
AKIHITO UCHIYAMA
10. R. Macias and C. Segovia, A decomposition into atoms of distributions on spaces of
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14. A. Uchiyama, On the compactness of operators of Hankel type, Tόhoku Math. J., 30
(1978), 163-171.
Received October 1, 1979.
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Vol. 92, No. 2
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Bruce Allem Anderson and Philip A. Leonard, Sequencings and Howell
designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Kevin T. Andrews, Representation of compact and weakly compact
operators on the space of Bochner integrable functions . . . . . . . . . . . . . . . . 257
James Glenn Brookshear, On the structure of hyper-real z-ultrafilters . . . . . . 269
Frank John Forelli, Jr., A necessary condition on the extreme points of a
class of holomorphic functions. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Richard J. Friedlander, Basil Gordon and Peter Tannenbaum, Partitions
of groups and complete mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Emden Robert Gansner, Matrix correspondences of plane partitions . . . . . . . 295
David Andrew Gay and William Yslas Vélez, The torsion group of a
radical extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
André (Piotrowsky) De Korvin and C. E. Roberts, Convergence theorems
for some scalar valued integrals when the measure is Nemytskii . . . . . . . . 329
Takaŝi Kusano and Manabu Naito, Oscillation criteria for general linear
ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
Vo Thanh Liem, Homotopy dimension of some orbit spaces . . . . . . . . . . . . . . . 357
Mark Mahowald, bo-resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .365
Jan van Mill and Marcel Lodewijk Johanna van de Vel, Subbases, convex
sets, and hyperspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
John F. Morrison, Approximations to real algebraic numbers by algebraic
numbers of smaller degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
Caroline Series, An application of groupoid cohomology . . . . . . . . . . . . . . . . . . 415
Peter Frederick Stiller, Monodromy and invariants of elliptic surfaces . . . . . 433
Akihito Uchiyama, The factorization of H p on the space of homogeneous
type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
Warren James Wong, Maps on simple algebras preserving zero products.
II. Lie algebras of linear type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469